Problem 15 - A Graphical Approach to College Algebra

Question

Skills Graph each piecewise-defined function in Exercises \(9-20 .\) Is \(f\) continuous on its domain? Do not use a calculator. $$f(x)=\left\\{\begin{array}{ll} -\frac{1}{2} x^{2}+2 & \text { if } x \leq 2 \\\ \frac{1}{2} x & \text { if } x>2 \end{array}\right.$$

Step-by-Step Solution

Answer

The function is not continuous at \( x = 2 \) because the left and right limits differ.

Step 1: Identify the Function Pieces

We have a piecewise-defined function with two parts: 1. For \( x \leq 2 \), the function is \( f(x) = -\frac{1}{2}x^2 + 2 \).2. For \( x > 2 \), the function is \( f(x) = \frac{1}{2}x \).

Step 2: Sketch the First Piece of the Function

For \( x \leq 2 \), we need to plot \( f(x) = -\frac{1}{2}x^2 + 2 \). This is a downward-opening parabola. Find the points by substituting values:- When \( x = 0 \), \( f(0) = 2 \)- When \( x = 2 \), \( f(2) = -\frac{1}{2}(2^2) + 2 = 0 \)Connect these points with a curve that extends to the left.

Step 3: Sketch the Second Piece of the Function

For \( x > 2 \), we need to plot \( f(x) = \frac{1}{2}x \). This is a straight line. Find the value for a few points:- When \( x = 2 \), \( f(2) = 1 \)- When \( x = 3 \), \( f(3) = \frac{3}{2} = 1.5 \)Continue the line to the right, starting from \( x > 2 \).

Step 4: Check Continuity at the Transition Point \( x = 2 \)

To check for continuity at \( x = 2 \), calculate the left-hand limit and right-hand limit:- Left limit as \( x \to 2^{-} \) using \( f(x) = -\frac{1}{2}x^2 + 2 \): \( f(2) = 0 \).- Right limit as \( x \to 2^{+} \) using \( f(x) = \frac{1}{2}x \): \( f(2) = 1 \).Since the left limit does not equal the right limit, \( f(x) \) is not continuous at \( x = 2 \).

Step 5: Conclusion on Continuity Across the Domain

Since the function is not continuous at \( x = 2 \) due to differing left and right limits, the function is not continuous over its entire domain.

Key Concepts

Continuity

Continuity in mathematical functions is a fundamental concept that tells us if a function, like our piecewise-defined one, is smooth or not. Imagine drawing the graph of a function without lifting your pen—that's continuity! For a function to be continuous at a point, three conditions must be satisfied:

• The function must be defined at that point.
• The limit of the function as it approaches the point from the left and from the right must exist.
• The left-hand limit and the right-hand limit must be equal to the function value at that point.

In our example, at the point where the function changes (at \( x = 2 \)), we calculated both the left-hand and right-hand limits. The piece where \( x \leq 2 \) gives us \( f(2) = 0 \), while the piece where \( x > 2 \) gives us \( f(2) = 1 \). Because they are not equal, the function shows a jump at \( x = 2 \), so it's not continuous there.

Graphing Functions

Graphing functions provides a visual representation of how a function behaves across its domain. With piecewise functions, it involves plotting each segment of the function separately.
For our piecewise-defined function, we have two segments:

• A parabola, \( f(x) = -\frac{1}{2}x^2 + 2 \) for \( x \leq 2 \), which opens downward.
• A straight line, \( f(x) = \frac{1}{2}x \) for \( x > 2 \).

Start by sketching each section according to its defined interval:- Use test points, like \( x = 0 \) and \( x = 2 \) for the parabola to find locations of pivotal points.- Similarly, use points like \( x = 2 \) and \( x = 3 \) for the line, to ensure accuracy.Remember, where the pieces meet, the continuity may change, which must be checked separately.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of several terms in the form of \( ax^n \), where \( n \) is a non-negative integer. These functions can be simple lines or complex curves, depending on the degree of the polynomial.
In our example, the first part of the piecewise function is a quadratic polynomial, \( f(x) = -\frac{1}{2}x^2 + 2 \). This makes a parabola and is characterized by:

• A degree of 2, indicating it's a curve rather than a line.
• An opening direction pointing downward because of the negative coefficient \(-\frac{1}{2}\).

The second part of our piecewise function, \( f(x) = \frac{1}{2}x \), is a linear polynomial:

• A degree of 1, indicating it's a straight line.
• A slightly upward slope given by the positive slope coefficient \( \frac{1}{2} \).

Domain of a Function

The domain of a function refers to all possible input values (or \( x \)-values) for which the function is defined. It gives a sense of the 'reach' of a function on the \( x \)-axis.
For our piecewise function, the domain can be discussed in parts, as different rules apply:

• The segment \( f(x) = -\frac{1}{2}x^2 + 2 \) is for \( x \leq 2 \), indicating that in this rule, \( x \) includes values up to and including 2.
• For \( x > 2 \), the segment \( f(x) = \frac{1}{2}x \) applies, covering \( x \) values greater than 2.

Overall, the piecewise function's domain can be expressed as the interval \( (-\infty, \infty) \), because it covers all real numbers. Just remember, at \( x = 2 \), the function's continuity is interrupted.